Globally symmetric Riemannian space

A Riemannian manifold each point of which is an isolated fixed point of some involutory isometry of , i.e. is the identity transformation. Let be the component of the identity in the group of isometries of the space and let be the isotropy subgroup at the point . Then is the homogeneous space , and the mapping is an involutory automorphism of ; moreover, is contained in the closed subgroup of all fixed points of and contains the component of the identity in .

Let be a real Lie algebra, let be an involutory automorphism of it and let be the subalgebra in of all -fixed elements. Consider the connected subgroup of the associated group corresponding to the subalgebra . If the group is compact, then is called a compactly-imbedded subalgebra of , while the pair is known as an orthogonal symmetric Lie algebra. Let be the decomposition into the eigen subspaces of corresponding to the eigen values 1 and . The pair is known as: a) an algebra of compact type if is compact and semi-simple; b) an algebra of non-compact type if is a Cartan decomposition; and c) an algebra of Euclidean type if is an Abelian ideal in . Let be an orthogonal symmetric Lie algebra and let be the above decomposition. Denote by the subset of the complex hull of . The mapping

is an involutory automorphism of the algebra , and is an orthogonal symmetric Lie algebra, called the dual to . If is an algebra of compact type, then is an algebra of non-compact type and vice versa.

Each globally symmetric Riemannian space generates an orthogonal symmetric Lie algebra , where is the Lie algebra of the group and ( is the identity in the group). is called a space of compact or non-compact type, depending on the type of the pair it generates. All simply-connected globally symmetric Riemannian spaces are direct products: , where is a Euclidean space, and and are globally symmetric Riemannian spaces of compact and non-compact types, respectively. For any space of non-compact type the curvature is non-positive in any two-dimensional direction; this curvature is everywhere non-negative for spaces of compact type. Any space of non-compact type is diffeomorphic to a Euclidean space.

Let be a globally symmetric Riemannian space of compact or non-compact type. The rank of is the maximal dimension of a flat totally-geodesic submanifold in . Let and be two flat totally-geodesic submanifolds of of dimension , let , , and let be the tangent vector to at the point . In such a case: 1) there exists an element such that and ; and 2) there exists an element such that and is the tangent vector to at .

Let be an orthogonal symmetric Lie algebra and let and be the eigen subspaces of corresponding to the eigen values 1 and . The algebra is called irreducible if the following conditions are satisfied: 1) is a semi-simple algebra and does not contain non-zero ideals of ; and 2) the algebra acts irreducibly on . A globally symmetric Riemannian space is called irreducible if the orthogonal symmetric Lie algebra generated by is irreducible. Two orthogonal symmetric Lie algebras and are called isomorphic if there exists an isomorphism of the algebra onto such that . The classification of simply-connected irreducible globally symmetric Riemannian spaces up to an isometry is equivalent to the classification of irreducible orthogonal symmetric Lie algebras up to an isomorphism.

The irreducible orthogonal symmetric Lie algebras of compact type are: I. , where is a compact simple Lie algebra and is any of its involutory automorphisms; II. , where the compact algebra is a direct sum of two simple ideals that are conjugate through the automorphism .

The irreducible orthogonal symmetric Lie algebras of non-compact type are: III. , where is a simple non-compact Lie algebra over whose complex hull is a simple Lie algebra over and is an involutory automorphism of such that its fixed points constitute a maximal compactly-imbedded subalgebra; IV. , where is a simple Lie algebra over considered as a real Lie algebra and is the conjugation of with respect to a maximal compactly-imbedded subalgebra , i.e. a mapping , . Moreover, if denotes the algebra dual to , then the latter is of type III or IV if is, respectively, of type I or II, and vice versa.

Only one globally symmetric Riemannian space is connected with each irreducible orthogonal symmetric algebra of non-compact type, and this space is simply connected. As regards compact algebras, the solution of the corresponding problem is much more complicated. It is sufficient to consider globally symmetric Riemannian spaces of type I and type II, connected with algebras of type II — these are in fact connected compact simple Lie groups endowed with a Riemannian structure which is invariant under left and right translations. The problem of classifying globally symmetric Riemannian spaces connected with algebras of type I, up to local isometries, is equivalent to the problem of classifying involutory automorphisms of simple compact Lie algebras. The global classification of the symmetric Riemannian spaces connected with a given orthogonal symmetric algebra of compact type is solved by the following theorem.

Let be an orthogonal symmetric algebra of compact type, where the subalgebra of fixed points of contains no ideals of other than . Let be a simply-connected Lie group with Lie algebra , let be the centre of , let be an automorphism of such that , and let be the set of fixed points of . For an arbitrary subgroup of one puts . The globally symmetric Riemannian spaces connected with coincide with the spaces of the form where , , endowed with an arbitrary -invariant metric. Here runs through all subgroups of , while runs through all subgroups of for which .

References

[1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2]	O. Loos, "Symmetric spaces" , 1–2 , Benjamin (1969)